Strange, charmed and b-flavoured mesons in an effective power-law potential

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  • Volume lOlB, number 4 PHYSICS LETTERS 14 May 1981


    N. BARIK and S.N. JENA Department ofphysics, Utkal University, Bhubaneswar-751004, Orissa, India

    Received 12 January 1981

    We have shown that an effective non-coulombic power-law potential, generating spindependence through scaler and vector exchanges in almost equal proportions along with zero quarkanomalous moment, which was found successful in earlier works for the fine-hyperfine splittings of heavy mesons like $ and T, can also describe very satisfactorily the S-wave hyperfine levels, Ml -transition rates and electromagnetic mass difference of the strange, charmed and b-flavoured mesons.

    It has been shown recently [l-3] that a purely phe- nomenological approach to the potential-model study of $ and T-spectra can as well lead to a static non- coulombic power-law potential of the form, V(r) = UT, with u close to 0.1. Although this phenome- nological potential gives quark confinement (a, u > 0) at long distances, its short-distance non-singular behav- iour is in apparent contradiction with what one ex- pects from Quantum Chromo Dynamics (QCD). How- ever, such non-singular short-distance behaviour does not pose any problem in understanding the hyperfine splitting (where it is believed to play an important role) of the heavy-meson spectra if we take this power-law potential V(r) = ar t V. to be an approximately equal admixture of vector and scalar components [3]. In that case one would have serious doubts in saying that the charmonium and upsilon spectra suggest a short-range Coulomb-like behaviour of the quark-anti- quark potential well in accordance with QCD predic- tions. Although this is an open question, still it needs to be tested in other areas of meson-spectroscopy, par- ticularly with the heavier quarkonium spectrum of tf. But in the absence of any evidence yet for these heavier mesons upto an energy scale of 35.8 GeV [4], we choose to test the applicability of this effective power-law potential with the same Lorentz-Dirac

    1 Work supported in part by the University Grants Commis- sion, India under the Faculty Improvement Programme.

    character as found in ref. [3] in the study of other light and heavy mesons in a unified manner.

    For convenience we select a group of light and heavy mesons which are non-selfconjugate in their quark constituents and study their ground-state hyper- fine mass splittings, radiative decay widths and finally the electromagnetic mass differences (EMD). The quark-antiquark configurations qIq2 like dii, (sd, SE), (cS, cd, cU), (b- b s, d, bti) and bZ would correspond to the non-selfconjugate meson states (n- , p-), (K*O, K, K*-, K-), (FE+, F,+, D,*+,D;, DE, Dg), (F;o, F{ Dg, D{, D{-, DC) and E-, respectively. All theie mesons excepting the b-flavoured mesons have been experimentally observed. However, with PETRA and PEP in operation now, the heavier b-flavoured me- sons are expected to be discovered in the near future. Therefore a study of the hyperfine mass splittings and electromagnetic mass differences of these mesons will be crucial in fixing their decay mechanism as well as in determining their production ratios in the e+e--an- nihilation experiments.

    We take the static potential in the form

    Vq,~l(r) = V. t a+ , (1)

    where a and v are positive. A solution to the Schriidinger equation with this potential would lead to the spin-averaged masses Mo(ql q2) of the mesons with quark-antiquark configuration q1 qZ. But the quanti- tative explanation of the hyperfine level depends on

    282 0 03 l-9 163/8 1 /OOOO-0000/S 02.50 0 North-Holland Publishing Company

  • Volume lOlB, number 4 PHYSICS LETTERS 14 May 1981

    the spin structure of the static potential. If we follow the same prescription as in ref. [3], we should regard the power-law potential to be an admixture of vector and scalar components with a vector fraction n. Then with zero quark anomalous moment, the relevant spin-dependent correction term generated by this po- tential in the usual manner for the consideration of the ground S-states only, would be

    6 vspin(r) = A(r)S1 s2 3



    A(r) = (2Q/3mq1mg,)[u(u + l)QF2] . (3)

    Now using a first order perturbation approach for the spin-dependent correction term 6 Ispin( one can ob- tain the ground state hyperfine masses as

    M(q& 3S,) =&(q&) + &l(r))ls 9

    WI&, 1 so> =y)(q&) - :u(r)& . (4)

    Hence the knowledge of the expectation values U(r))ls would enable one to determine the hyperfine levels of various qlq2-systems in their ground state. In fact there should be spin-independent relativistic cor- rections present along with the spin-dependent terms. However, if we are interested only in the relative mass splittings and not in the absolute mass scale, then there would be no loss of generality in ignoring the spin-inde- pendent correction terms. Nevertheless we take a sim- plifying assumption that the effects due to spin-inde- pendent correction terms can, in some phenomenolo- gical sense, be taken into account by choosing a suit- able non-zero value for the parameter V. in the static potential (1) to obtain a good fit to the spin-averaged masses Mo(qI q2). Therefore we may expect to have different values of V. = Vo(qI q2) for different ql q2- systems, while keeping the parameters ?I and u to be the same for all systems in order to maintain an ap- parent flavour independence of the potential. Now to obtain the potential parameters Y and u and the quark mass parameters m,, me and mb , we make a spe- cial reference to the spin-averaged masses of the self- conjugate mesons 9, $ and I. With V. = -7.38 GeV, we obtain

    (a, u) = (6.08 GeV, 0.113))

    (m,,m,,q,) = (623.0, 1854.5,5215.0 MeV) . (5)

    Table 1 The spin averaged ground state massesMo(qrqa), the ground state expectation values (r u-a)ts, and the corresponding po- tential parameters Vu(qrq2) for various qrqa systems.

    Meson symbol


    (CeV) (r-2)ls Ilg(q142)


    P- 0.6101 0.2948 -7.483

    Ko 0.7958 0.3267 -7.455

    K- 0.7914 0.3217 -7.455

    DE 1.9735 0.3874 -7.397 0

    DC 1.9703 0.3820 -7.397

    PZ 2.115 0.5433 -7.35

    PbO 5.379 0.6345 -7.38

    @ 5.3106 0.4322 -7.38

    % 5.3071 0.4284 -7.38

    E- 6.287 1.4191 -7.38 _

    Then in order to obtain the quark masses md and mu, we fit the experimental spin-averaged masses of the DE(cd) and Dz(cii) systems with m,, II and u fixed as in (5). This gives with Vo(cd) = Vo(cti) = -7.397 GeV

    (md, mu) = (385.209,379.31 MeV) . (6)

    In this manner we fix the quark mass parameters mu, md, m,, m, and mb and the potential parameters a and u. Now with suitable values of vo(ql q2) we can obtain the spin-averaged ground-state masses of various qlq2-systems close to the corresponding experimental values. We present in table 1, the results for the spin- averaged ground-state masses Mo(ql q2), the ground- state expectation values (r-2)ls and the corresponding potential parameters Trg(qlq2). We find that the Vo(ql q2) for various q1 q2-systems are not very signi- ficantly different from each other since all these values he in a range -7.35 GeV to -7.483 GeV. This may be interpreted as an indication to the fact that the effect due to the relativistic spin-independent correction terms is not quite significant.

    Now using (ru-2)Is and the adjustable parameter Q in eq. (3) we can calculate the triplet- and singlet- state masses along with the hyperfine splitting (A (r)jIs from eqs. (4). We find that the vector fraction 4 = 0.58 gives a good agreement to the hyperfine masses of most of the experimentally observed mesons under considera- tion. The results obtained are compared with the corre- sponding experimental masses in table 2. We find the agreement better than what we can expect from such a


  • Volume lOlB, number 4 PHYSICS LETTERS 14 May 1981

    Table 2 Calculated hyperfine meson massesMof the light and charmed mesons compared with the corresponding experi- mental values.


    Meson M Experimental symbol (MeV) mass (MeV) ____

    P- 159.25 710 f 5

    n- 162.63 139.57 f 0.01

    is*0 896.41 896 f 1.0 K0 493.96 491.1 f 0.2

    K*- 892.04 892 f 1.0 K- 489.47 493.61 f 0.04

    I$- 2149.70 2140.0 f 60

    FE 2010.70 2030 * 60

    *+ DC 2013.60 2008.6 f 1.0

    DZ 1853.20 1868.4 ? 1.9

    Dc*O 2010.40 2006 * 1.5 0

    DC 1849.80 1863.3 f 0.9

    simplified non-relativistic potential-model approach. This then supports our contention that the static con- fining potential generates the spin dependence in a uni- fied manner as an approximately equal admixture of vector and scalar parts. We then predict the hyperfine levels for the b-flavoured mesons as follows.

    A4(Fgj = 5.3934 GeV , M(Ftj = 5.3356 GeV ,

    M(~*Oj = 5.3265 GeV, M(Dtj = 5.2628 GeV ,

    IIf = 5.3231 GeV, M(x) = 5.2590 GeV , (7)

    M(E*-j = 6.2978 GeV, M(E-j = 6.2544 GeV .

    These results agree very well with the preliminary evi- dence obtained by an experiment at CERN-SPS [5], showing an enhancement at 5.3 GeV in the mass spec- trum of the $ + K t 71 combinations produced by high- energy pion beams. Here it is important to note that for the b-flavoured mesons the difference in the pre- dicted masses of the triplet and singlet states is much less than a pion mass. Hence for these mesons the dom- inant decay modes would be radiative ones. In view of the fact that the radiative decay is the most important decay mode for heavier mesons, we calculate the Ml- transition rates using the naive quark model formula



    rY3s, +'hJ+Y)

    =:oLp3(e4,12mqI +eq2/2mq,j2, (8)

    where eql and eq2 are the charges of the constituent quarks and P photon momentum given by

    P= [Mq3s,j -M2($))]/2M(%~j. (9)

    The results obtained are as follows:

    l?(D;+ -+Dz tyj=2.27 keV,

    I(D,* + Di t yj = 0.53 keV ,

    I(D; +Dz+7)=39.89keV,

    r(Dc-+Di tyj= 1.8keV,

    r(F,*+ --f Fc tyj=O.l8keV,

    r(F~-+F~ +yj=O.l7keV,

    r(E*-+E-+yj=O.O2keV. (10)

    Finally, it would be interesting to obtain the so- called electromagnetic-mass differences (EMD) of the strange (Kj, charmed (DC) and b-flavoured (Dbj me- sons. Due to their importance in the kinematics of production and decay, the EMD-values for charmed and b-flavoured mesons have drawn considerable atten- tion in recent years [7]. The contributions to EMD are believed to come from (ij the differences in the spin- averaged masses (ii) the possible differences in the spin-spin interaction 6 I/spin(rj arising from the strong- interaction gluon exchange and (iii) the difference in the single-photon exchange contribution in the lowest order given by the Coulomb and the magnetic dipole interaction terms

    I/photon (rj = ae e 91 q2

    X {l/r - [8n6(3)(rj/3mqlmq2]S1 *S2} . (11)

    We have discussed in ref. [8] that in a potential-model study where the parameters are usually obtained from an initial fit to the experimental spin-averaged masses, the coulombic contribution part due to the photon ex- change is in a way phenomenologically taken into ac- count. Then avoiding the possible double counting due to this coulombic contribution part the EMD-values for the strange (Kj, charmed (DC) and b-flavoured me- sons can be written [8] in the following manner.

  • Volume lOlB, number 4 PHYSICS LETTERS 14 May 1981

    (i) K-mesons:

    (K*O _ K*-) =M(K*o)-M(K*-)

    + (2no/27m,)R, I Jl,(O> Ii ,

    (it- K-) = M&O) - M(K-)

    - (2no/9m,)R, I ti,(O) 1; .

    (ii) Dc-mesons:


    (DE - DE) = M(D;+) - M(D;O)

    - (4~o/27m,)R, I J/,(O) I b, 9 (D; - D;) = M(D;) - M(D;)

    + (4no/9m,)R, I J/,(O) I& .

    (iii) Db-mesons:


    (D; - D;-) =M(D;) -M(q-)

    + (2~o/27m&,, I d,(o) I;, ,

    (Dt - DC) =M(D;) -M(D;)

    - (2flo/9m&,, I d,(o) lkb . (14)

    Here we have taken M(%!*) and M(w) to be the triplet- and singlet-state hyperfine masses, respectively, of a meson 3fI as given in eq. (4) and R, = (1 /md t 2/m,). Again the I J/,(O) I& as obtained in the pro- cess of the numerical computation are as follows:

    0.01797,0.02127 GeV3} . (15)

    Then the EMD-values for strange and charmed mesons calculated from eqs. (12) and (13) can be listed along with the corresponding experimental values in the fol- lowing manner :

    (iT*O -K*-) = 4.62 MeV, Expt.(4.1 + 0.6)MeV,

    (go - K-) = 3.73 MeV, Expt.(394 f O.l3)MeV,

    (D,*+ - DE) = 2.94 MeV, Expt.(2.6 f 1.8)MeV,

    (Dz - DE) = 4.18 MeV, . Expt.(5.0 ?r 0.8)MeV .


    These calculations show remarkably good agreement with the experimental EMD-values. Then our predic- tions of the electromagnetic mass differences of the

    b-flavoured mesons obtained from eq. (14) are as follows:

    (D; - q*-) = 3.45 MeV,

    (@ - DC) = 3.64 MeV . (17)

    It is interesting to note in this context that the result obtained in ref. [9] by using the extension of the Cottingham formula to SU(5) is of opposite sign with (DE - Db) = -2 MeV. However, a calculation in the MIT-bag model [lo] gives (q - D{-) = (1.54 f 0.8) MeV and (DE - Db) = (1.46 f 0.8) MeV. In any case future experiments would certainly verify these pre- dictions.

    Finally, we must point out that the average kinetic energy (T) computed for the meson states under con- sideration in this power-law potential model comes out to be of the order of 0.38 GeV. This near equality of (T) for all light and heavy mesons is reminiscent of a logarithmic potential which is a limiting case of the power-law potential. Therefore the above result is con- sistent with a very small value of u equal to 0.113 ob- tained in our analysis. Again the kinetic energy average (T) = 0.38 GeV would justify the non-relativistic Schrodinger approach to the bound-state problem even in the case of light mesons. Thus it is apparent that the relativistic effects which are expected to be important in the case of light mesons are found to be phenomeno- logically less significant for some unknown reasons. Such observations were also made by many other authors [ 1 l] while studying the meson and baryon spectra in non-relativistic potential models. Thus, in conclusion we point out that it is possible to study the light and heavy mesons in a unified manner at least in their ground states in a non-relativistic power-law potential-model.

    We are thankful to Professor B.B. Deo for his con- stant inspirations and valuable suggestions. We also thank the Computer Centre, Utkal University for its timely cooperation in the computational work. This work is supported in part by the University Grants Commission of India under the Faculty Improvement Programme.


  • Volume lOlB, number 4


    PHYSICS LETTERS 14 May 1981

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